Adiabatic progression with intermediate re-optimization to solve hard variational quantum problems in quantum computing

ABSTRACT

A hybrid classical-quantum computing device to execute a quantum circuit corresponding to a variational problem, is configured. The configuring further comprises causing the hybrid classical-quantum computing device to execute the quantum circuit by performing an adiabatic progression operation, wherein the adiabatic progression operation comprises increasing the difficulty of the variational problem from a simplified version of the problem to the variational problem.

RELATED APPLICATION

This application claims priority to U.S. Provisional Application Ser.No. 62/831,418, filed Apr. 9, 2019, titled, “Adiabatic Progression withIntermediate Re-Optimization to Solve Hard Variational Quantum Problemsin Quantum Computing”, which is hereby incorporated by reference in itsentirety.

TECHNICAL FIELD

The present invention relates generally to a method, system, andcomputer program product for operating a quantum computing dataprocessing environment to solve certain types of problems. Moreparticularly, the present invention relates to a method, system, andcomputer program product for adiabatic progression with intermediatere-optimization to solve hard variational quantum problems in quantumcomputing.

BACKGROUND

Hereinafter, a “Q” prefix in a word of phrase is indicative of areference of that word or phrase in a quantum computing context unlessexpressly distinguished where used.

Molecules and subatomic particles follow the laws of quantum mechanics,a branch of physics that explores how the physical world works at themost fundamental levels. At this level, particles behave in strangeways, taking on more than one state at the same time, and interactingwith other particles that are very far away. Quantum computing harnessesthese quantum phenomena to process information.

The computers we use today are known as classical computers (alsoreferred to herein as “conventional” computers or conventional nodes, or“CN”). A conventional computer uses a conventional processor fabricatedusing semiconductor materials and technology, a semiconductor memory,and a magnetic or solid-state storage device, in what is known as a VonNeumann architecture. Particularly, the processors in conventionalcomputers are binary processors, i.e., operating on binary datarepresented in 1 and 0.

A quantum processor (q-processor) uses the odd nature of entangled qubitdevices (compactly referred to herein as “qubit,” plural “qubits”) toperform computational tasks. In the particular realms where quantummechanics operates, particles of matter can exist in multiplestates—such as an “on” state, an “off” state, and both “on” and “off”states simultaneously. Where binary computing using semiconductorprocessors is limited to using just the on and off states (equivalent to1 and 0 in binary code), a quantum processor harnesses these quantumstates of matter to output signals that are usable in data computing.

Conventional computers encode information in bits. Each bit can take thevalue of 1 or 0. These 1s and 0s act as on/off switches that ultimatelydrive computer functions. Quantum computers, on the other hand, arebased on qubits, which operate according to two key principles ofquantum physics: superposition and entanglement. Superposition meansthat each qubit can represent both a 1 and a 0 at the same time.Entanglement means that qubits in a superposition can be correlated witheach other in a non-classical way; that is, the state of one (whether itis a 1 or a 0 or both) can depend on the state of another, and thatthere is more information that can be ascertained about the two qubitswhen they are entangled than when they are treated individually.

Using these two principles, qubits operate as more sophisticatedprocessors of information, enabling quantum computers to function inways that allow them to solve difficult problems that are intractableusing conventional computers. IBM has successfully constructed anddemonstrated the operability of a quantum processor usingsuperconducting qubits (IBM is a registered trademark of InternationalBusiness Machines corporation in the United States and in othercountries.)

A superconducting qubit includes a Josephson junction. A Josephsonjunction is formed by separating two thin-film superconducting metallayers by a non-superconducting material. When the metal in thesuperconducting layers is caused to become superconducting—e.g. byreducing the temperature of the metal to a specified cryogenictemperature—pairs of electrons can tunnel from one superconducting layerthrough the non-superconducting layer to the other superconductinglayer. In a qubit, the Josephson junction—which functions as adispersive nonlinear inductor—is electrically coupled in parallel withone or more capacitive devices forming a nonlinear microwave oscillator.The oscillator has a resonance/transition frequency determined by thevalue of the inductance and the capacitance in the qubit. Any referenceto the term “qubit” is a reference to a superconducting qubit oscillatorcircuitry that employs a Josephson junction, unless expresslydistinguished where used.

The information processed by qubits is carried or transmitted in theform of microwave signals/photons in the range of microwave frequencies.The microwave frequency of a qubit output is determined by the resonancefrequency of the qubit. The microwave signals are captured, processed,and analyzed to decipher the quantum information encoded therein. Areadout circuit is a circuit coupled with the qubit to capture, read,and measure the quantum state of the qubit. An output of the readoutcircuit is information usable by a q-processor to perform computations.

A superconducting qubit has two quantum states—|0> and |1>. These twostates may be two energy states of atoms, for example, the ground (Ig>)and first excited state (Ie>) of a superconducting artificial atom(superconducting qubit). Other examples include spin-up and spin-down ofthe nuclear or electronic spins, two positions of a crystalline defect,and two states of a quantum dot. Since the system is of a quantumnature, any combination of the two states are allowed and valid.

For quantum computing using qubits to be reliable, quantum circuits,e.g., the qubits themselves, the readout circuitry associated with thequbits, and other parts of the quantum processor, must not alter theenergy states of the qubit, such as by injecting or dissipating energy,in any significant manner or influence the relative phase between the|0> and |1> states of the qubit. This operational constraint on anycircuit that operates with quantum information necessitates specialconsiderations in fabricating semiconductor and superconductingstructures that are used in such circuits.

Quantum computing can often be used to solve problems more quickly thanin conventional computing. For example, one quantum algorithm isGrover's Search, which accomplishes searching through an unordered listof N items with fewer lookups than is the case in conventionalcomputing.

Quantum gates are the elementary building blocks for quantumcomputation, acting on qubits the way classical logic gates act on bits,one and two at a time, to change qubit states in a controllable way. AnX gate inverts the state of a single qubit, much like a NOT gate invertsthe state of a single bit in classical computing. An H gate, or Hadamardgate, puts a single qubit into a state of superposition, a combinationof the 0 and 1 quantum states. The qubit only resolves to a definitestate when measured. For example, when provided with an input having aquantum state of 0, within the Hadamard gate the quantum state is insuperposition, but the output has a 50 percent probability of being inthe quantum 0 state and a 50 percent probability of being in the quantum1 state. Other single-qubit gates alter the qubit state in other definedways.

Multi-qubit gates implement gates that perform conditional logic betweenqubits, meaning the state of one qubit depends on the state of another.For example, a Controlled-NOT, or CNOT gate, has two qubits, a targetqubit and a control qubit. If the control qubit is in the 1 quantumstate, the CNOT gate inverts the state of the target qubit. If thecontrol qubit is in the 0 quantum state, the CNOT gate does not changethe state of the target qubit.

Multiple qubits can also be entangled. Two or more qubits are entangledwhen, despite being too far apart to influence one another, they behavein ways that are individually random, but also too strongly correlatedto be explained by supposing that each object is independent from theother. As a result, the combined properties of an entangled multi-qubitsystem can be predicted, but the individual outcome of measuring eachindividual qubit in such a system cannot.

Similar to conventional computing, quantum computing gates can beassembled into larger groups, called quantum circuits, to perform morecomplicated operations. For example, a SWAP gate, which exchanges thestates of a pair of qubits, can be constructed from three CNOT gates.

Quantum circuits can perform some operations in parallel, and some inseries. The length of the longest series in the program is also referredto as the depth of the quantum circuit. For example, the three CNOTgates comprising a SWAP gate are arranged in series, giving a depth of3. Programs with a shallower depth take less execution time and providebetter performance, so are preferred.

Conventional computers do not have to be hand-programmed with specificinstruction steps, such as those provided in processor-specific assemblylanguages. Instead, programmers write hardware-independent code in ahigher-level language, and a compiler translates this code into assemblylanguage for execution on a specific processor. Similarly, in quantumcomputing programmers do not have to specify individual gates. Instead,programmers can write higher-level code in a higher-level language. Acompiler parses this code and maps it into a quantum circuit. Finally, aquantum processor executes the quantum circuit. Quantum programmers canalso make use of already-programmed libraries, for use in solvingproblems in areas such as chemistry, artificial intelligence, andoptimization.

A class of problems exists called variational problems, of which thereare subclasse—hard variational problems and easy variational problems.Optimization problem is a non-limiting example of variational problems,which can be hard or easy depending on the dimensionality, numerosity ofnodes, level of accuracy desired, and other factors. An optimizationproblem is a computational problem in which the best or optimal solutionis to be determined for a different problem where the different problemhas several possible solutions.

For example, the different problem can be the famous traveling salesmanproblem where a route has to be determined between several cities suchthat a traveling salesman covers each of the of cities without revisingany of the cities. This problem has many possible solutions—routesbetween the cities. An optimization problem related to the travelingsalesman problem is to find the shortest—i.e., the best or most optimalroute—from the many possible routes, each of which satisfies therequirements of the traveling salesman problem.

Another example of the different problem is the max cut problem. In agraph, solving the max cut problem means a subset S of the vertex setsuch that the number of edges between S and the complementary subset(the subset of vertices not in S) is as large as possible.

Computing a dissociation curve is another non-limiting example ofvariational problems, which can be hard or easy depending on thedimensionality, level of accuracy desired, and other factors. Adissociation curve plots, on a graph, an energy cost of pulling apartatoms in a molecule. The x-axis of the graph is the distance betweenatoms, and the y-axis of the graph is an amount of energy.

Configuring a variational problem for execution on a computer so thatthe computer can compute the optimal solution in polynomial time is adifficult problem in itself. Until recently, the only computingresources available for executing variational problems were theconventional computers as described herein. Many variational problemsare too difficult or too complex for conventional computers to computein polynomial time with reasonable resources. Generally, an approximatedsolution which can be computed in reasonable time and with reasonableresources is accepted as the near-optimal solution in such cases.

The advent of quantum computing has presented advancement possibilitiesin many areas of computing, including the computation of variationalproblems. Because a quantum computing system can evaluate many solutionsfrom the solution space at once, the illustrative embodiments recognizethat such systems are particularly suitable for solving variationalproblems.

Variational quantum algorithms use classical optimizers to search ahigh-dimensional non-convex parameter space for a quantum circuitsolving a particular problem. In general, optimization inhigh-dimensional non-convex spaces is very hard, and convergence to asolution becomes increasingly difficult as (meaningful) dimensionalityincreases. For quantum variational problems, this means it will be verydifficult, if not impossible, to solve most problems requiringoptimization in high-dimensional non-convex spaces with the presenttechnology. State-of-the-art chemistry or combinatorial optimizationproblems can require thousands of variational parameters to dependablyconverge, while modern methods usually struggle to converge with only afew dozens of such parameters. The system that is the subject of thepresent invention has allowed us to improve the prospects of convergencefor chemistry and optimization problems, where convergence to a desireddegree of accuracy was previously impossible. Related topics includebootstrapping for improving dissociation curve convergence, gate-basedquantum adiabatic methods, homotopy methods for convex optimization, andinterior point methods for optimization.

SUMMARY

The illustrative embodiments provide a method, system, and computerprogram product. An embodiment includes a method for adiabaticprogression with intermediate re-optimization to solve hard variationalquantum problems in quantum computing. An embodiment configures a hybridclassical-quantum computing device to execute a quantum circuitcorresponding to a variational problem. The configuring furthercomprises causing the hybrid classical-quantum computing device toexecute the quantum circuit by performing an adiabatic progressionoperation, wherein the adiabatic progression operation comprisesincreasing the difficulty of the variational problem from a simplifiedversion of the problem to the variational problem.

An embodiment includes a computer usable program product. The computerusable program product includes a computer-readable storage device, andprogram instructions stored on the storage device.

An embodiment includes a computer system. The computer system includes aprocessor, a computer-readable memory, and a computer-readable storagedevice, and program instructions stored on the storage device forexecution by the processor via the memory.

BRIEF DESCRIPTION OF THE DRAWINGS

Certain novel features believed characteristic of the invention are setforth in the appended claims. The invention itself, however, as well asa preferred mode of use, further objectives and advantages thereof, willbest be understood by reference to the following detailed description ofthe illustrative embodiments when read in conjunction with theaccompanying drawings, wherein:

FIG. 1 depicts a block diagram of a network of data processing systemsin which illustrative embodiments may be implemented;

FIG. 2 depicts a block diagram of a data processing system in whichillustrative embodiments may be implemented.

FIG. 3 depicts a block diagram of an example configuration for adiabaticprogression with intermediate re-optimization to solve hard variationalquantum problems in quantum computing, in accordance with anillustrative embodiment;

FIG. 4 depicts a block diagram of an example configuration for adiabaticprogression with intermediate re-optimization to solve hard variationalquantum problems in quantum computing, in accordance with anillustrative embodiment;

FIG. 5 depicts an example of a hard variational quantum problem suitablefor solving using adiabatic progression with intermediatere-optimization to solve hard variational quantum problems in quantumcomputing, in accordance with an illustrative embodiment;

FIG. 6 depicts an example of solving a max cut problem using adiabaticprogression with intermediate re-optimization, in accordance with anillustrative embodiment;

FIG. 7 depicts an example of computing a dissociation curve usingadiabatic progression with intermediate re-optimization, in accordancewith an illustrative embodiment;

FIG. 8 depicts another example of computing a dissociation curve usingadiabatic progression with intermediate re-optimization, in accordancewith an illustrative embodiment;

FIG. 9 depicts a flowchart of an example process for adiabaticprogression with intermediate re-optimization to solve hard variationalquantum problems in quantum computing in accordance with an illustrativeembodiment; and

FIG. 10 depicts a flowchart of another example process for adiabaticprogression with intermediate re-optimization to solve hard variationalquantum problems in quantum computing in accordance with an illustrativeembodiment.

DETAILED DESCRIPTION

Computing of variational problems is a well-recognized technologicalfield of endeavor. Quantum computing using processors formed fromquantum qubits is another well recognized technological field ofendeavor. The present state of the technology in a combination of thesetwo fields of endeavor has certain drawbacks and limitations. Theoperations and/or configurations of the illustrative embodiments impartadditional or new capabilities to improve the existing technology inthese technological fields of endeavor, especially in adiabaticprogression with intermediate re-optimization to solve hard variationalquantum problems in quantum computing.

The illustrative embodiments recognize that solving a problem, such asan optimization problem, in quantum computing typically requirestranslating the optimization problem, along with its inputs, into anIsing Hamiltonian, and then passing the Ising Hamiltonian to a quantumvariational algorithm, such as the Variational Quantum Eigensolver (VQE)algorithm and the Quantum Approximate Optimization Algorithm (QAOA).

Ising (Z. Physik, 31, 253, 1925) introduced a model consisting of alattice of “spin” variables si, which can only take the values +1 (↑)and −1(↓). Every spin interacts with its nearest neighbors (2 in 1D) aswell as with an external magnetic field h. See an example 1-dimensionalIsing model, FIG. 3.

The Hamiltonian of the example Ising model is

${H\left( \left\{ s_{i} \right\} \right)} = {{{- J}{\sum\limits_{({i,j})}\;{s_{i}s_{j}}}} - {h{\sum\limits_{i}\; s_{i}}}}$

The sum <i, j> is over nearest neighbors (j=i±1 in 1D).

J is a constant specifying the strength of interaction. The term “spin”and “magnetic field” in the Ising model originate from its initialapplication to the phenomenon of spontaneous magnetization inferromagnetic materials such as iron. Each iron atom has an unpairedelectron and hence a net spin (or magnetic moment). At low temperature,the spins spontaneously align giving rise to a non-zero macroscopicmagnetic moment. The macroscopic magnetic moment disappears when thetemperature exceeds the Curie temperature (1043 K for iron).

The Ising model can be applied to many other problems beyond magnetism,such as phase separation in binary alloys, crystal growth, and solvingoptimization problems. Higher dimension Ising models are generally usedin solving many problems.

2D Ising model defined over a square lattice of N spins under periodicboundary conditions. Again, the Hamiltonian can be written as

${H\left( \left\{ s_{i} \right\} \right)} = {{{- J}{\sum\limits_{\langle{i,j}\rangle}\;{s_{i}s_{j}}}} - {h{\sum\limits_{i}\; s_{i}}}}$

J describes the strength of interaction, h is external magnetic field,and the sum is over all <i,j> nearest neighbor pairs. Each spin has 4nearest neighbors.

Optimization and other variational problems are computation-intensivetasks. A thorough analysis of a solution space of an optimization orother variational problem can easily take several years on acommercially available conventional computer. Therefore, theillustrative embodiments also recognize that computing the optimalsolution in a time-efficient manner using quantum computers is even moredifficult using the presently available methods.

The illustrative embodiments solve a hard variational problem to adesired degree of accuracy by beginning with a simplified version of theproblem, which can be solved to the user's degree of accuracy, and bythen carefully increasing the complexity of the problem while solvingeach more complex problem along the way, until finally the desiredproblem is solved. Increasing the complexity of the problem, whilesolving each more complex problem along the way, is also referred toherein as adiabatic progression. As an example, in the case ofdissociation curves in chemistry, this can mean starting with an easysection of the dissociation curve and progressing the Hamiltonian towardharder, more excited sections of the curve. As another example, inoptimization problems in which the target system is represented as agraph, it can mean beginning with only a few nodes or components to theproblem, and then gradually increasing the number of singular points.

Critically, the illustrative embodiments provide methods to control thepace at which to progress the difficulty of the problem, which is theprimary challenge here (attempting to solve the problem directly can beseen as poor pacing—jumping from the easy problem to the full problem inone step). If a step is paced too aggressively, the starting point ofthe intermediate optimization may jump out of the basin of attraction,which includes the desired solution, making the optimizer much lesslikely to find the solution, or causing it to take much longer toconverge.

An illustrative embodiment controls this pacing to ensure thatintermediate solutions reach the user's desired level of accuracy. Someexample problems with the current technology, and improvements of theillustrative embodiments over the existing technology are as follows.First, a method of the illustrative embodiments has been used with the“rewind” method below to complete dissociation curves and maxcutproblems to degrees of accuracy which previously were impossible bysolving with the variational algorithm directly, or via “bootstrapping”(a form of extrapolation with window depth equal to 1). Second, theimprovement over using classical homotopy or interior-point methods isthat an embodiment is able to use a quantum solution—the wave functionsof the intermediate solutions can maintain exponentially manyprobabilities compared to the size of the problem. Lastly, in the nearfuture, quantum hardware will be large enough that we will be unable toverify classically the solutions of quantum algorithms, and the issue ofincreasing the dependability of our quantum computations is pressing.

The illustrative embodiments recognize that the presently availabletools or solutions do not address these needs or provide adequatesolutions for these needs. The illustrative embodiments used to describethe invention generally address and solve the above-described problemsand other related problems by adiabatic progression with intermediatere-optimization to solve hard variational quantum problems in quantumcomputing.

An embodiment can be implemented as a software application. Theapplication implementing an embodiment, or one or more componentsthereof, can be configured as a modification of an existingquantum-classical hybrid data processing system—i.e., a nativeapplication in the classical computing system that produces inputs for aquantum computing system, as an application executing in a classicaldata processing system communicating with an existing quantum computingsystem over a network, as a separate application that operates inconjunction with an existing quantum-classical system in other ways, astandalone application for execution on a classical system, or somecombination thereof.

An embodiment configures a hybrid data processing environment includinga classical computing environment and a quantum computing environment.In accordance with the illustrative embodiments, the environmentincludes at least one quantum compute node (QCN), and at least oneconventional node (CN) on which an embodiment can execute. Such acomputing environment is hereinafter referred to as a quantum computingenvironment (QCE). The QCE may include one or more CNs in a suitableconfiguration—such as a cluster—to execute applications usingconventional binary computing. The hybrid environment can be, but neednot necessarily be implemented using a cloud computing architecture. Tosolve a given problem, an embodiment produces, in the classicalcomputing environment, a quantum circuit that is executed using thequantum computing environment.

An embodiment works as follows: when supplied a parametrized problem,with a single continuous parameter z, with the property that the problemcorresponding to z=0 is easy to solve, and the problem corresponding toz=1 (or some higher, user specified z) is the target (potentiallydifficult) problem that an embodiment is to solve. For example, if theproblem to solve is an optimization on a graph (such as max cut), theparametrized problem could control the number of edges in the graph,where with z=0 the graph has one edge (and the problem is easy tosolve), and with z=1 the graph has all the edges in the original probleminstance. In one embodiment, a smooth progression from the easy problem(z=0) to the hard problem (z=1) is available (this is studied in detailin the fields of homotopy optimization and interior point methods).

An embodiment starts with the known easy problem. The easy problem has acorresponding continuous parameter z. The continuous parameter z has acorresponding set of variational parameters. The set of variationalparameters defines a configuration of a quantum circuit that can be usedto solve the problem. Thus, to increase the complexity of the problem,an embodiment increases the continuous parameter z by an amount, thuscorrespondingly changing the set of variational parameters. The changedset of variational parameters defines a new configuration of a quantumcircuit, which can be used to attempt to solve the more-complex problem.

An embodiment can follow one of many methods, including one or both of arewind embodiment and a race embodiment, to perform the complexityincreases. Each method performs the complexity increases differently.

A rewind embodiment is suitable for use when a known correct solution tothe problem can also be obtained by another means. A rewind embodimentstarts with the continuous parameter z in the easy state, and increasesz to the goal state (i.e. the maximum z). A rewind embodiment generatesa set of variational parameters corresponding to the new z. A rewindembodiment then uses the quantum circuit configuration defined by theset of variational parameters to attempt to solve the problem. A rewindembodiment compares the output of the quantum circuit to a known correctsolution to the problem obtained by another means (such as by using someform of polynomial-time validation of the solution, or by comparing toan exact classical solution). If the output of the quantum circuit isbelow a threshold difference from the known-correct solution, theembodiment has solved the hard problem, meeting the goal. If, instead,the output of the quantum circuit is above a threshold difference fromthe known-correct solution, or the quantum circuit fails to completeexecution and produce a solution within a predetermined time period, therewind embodiment concludes that the increase in z has been too large.

If the increase in z has been too large, a rewind embodiment selects asmaller increase in z from the starting point. In one embodiment, thesmaller increase in z is half the size of the previous increase. Inanother embodiment, the smaller increase in z is ⅓. In anotherembodiment, the smaller increase is 1/n multiplied by the size of theprevious increase, where n is any positive whole number greater than 1.In another embodiment, the smaller increase can be selected usinganother suitable method.

Once a rewind embodiment has determined a smaller increase, theembodiment repeats the process of generating a set of variationalparameters corresponding to the new z, then using the quantum circuitconfiguration defined by the new set of variational parameters toattempt to solve the problem. If the new z succeeds in solving theproblem, the embodiment makes the new z the starting point, and attemptsto increase z in a manner described herein.

If the new z does not succeed in solving the problem, a rewindembodiment selects an even smaller increase in z from the startingpoint. In one embodiment, the even smaller increase is 1/n multiplied bythe size of the previous increase, where n is any positive whole numbergreater than 1. In another embodiment, the smaller increase can beselected using another suitable method. Once a rewind embodiment hasdetermined the even smaller increase, the embodiment repeats the processof generating a set of variational parameters corresponding to the newz, then using the quantum circuit configuration defined by the new setof variational parameters to attempt to solve the problem.

A rewind embodiment repeats the process until it arrives at an increasein z from the starting point that does solve the problem. The embodimentmakes the new z the starting point, and attempts to increase z onceagain in a manner described herein. Thus, a rewind embodiment proceedsto increase z as much as possible in each step until the embodimentgenerates a correct quantum circuit configuration with a goal value ofz.

Thus, if each increase is smaller than a previous increase, theoperations of a rewind embodiment can be understood as performing asearch (e.g. binary search, when each increase is one-half of theprevious increase) for the next point which lies inside the basin ofattraction when using the predecessor's variational parameters. Thissearch is limited by the possibility of rewinding infinitely and nevermaking progress, so in practice the search preferably should choose notto see or use some points very close to the predecessor.

A race embodiment is suitable for use if no method of validation isavailable. A race embodiment uses one or more pseudo-validation methodsto select one of several possible successor points. For example, in onerace embodiment, the difference in time to converge to an optimal pointwith a gradient-based optimizer when inside vs. outside the basin ofattraction is used as a pseudo-condition on progressing. Meaning, if theembodiment determines that points inside the basin of attractionconverge very quickly, and points outside the basin take 10× longer,then the embodiment can “race” the executions of these points againstone another to choose the next point.

Thus, a race embodiment starts with the continuous parameter z in theeasy state, and generates a set of increases in z from the startingpoint. In one embodiment, each of the set of increases is a fixeddistance from the previous increase, dividing a range from the startingpoint to the goal state into equally-sized segments. Another embodimentuses equally-sized segments, but only up to an intermediate stateinstead of the goal state. Another embodiment uses unequally-sizedsegments, selected based on known characteristics of the problem to besolved to maximize a chance at arriving at a solvable state of theproblem. For example, the increases could be smaller near one value of zand larger near another value of z. Another embodiment uses anothersuitable method of generating the set of increases.

For each new z in the set, a race embodiment generates a set ofvariational parameters corresponding to each new z. A race embodimentthen uses the quantum circuit configuration defined by each set ofvariational parameters to attempt to solve the problem.

Each of the quantum circuit configurations executes in parallel. Becauseexecution times of the set of quantum circuits tend to cluster, a raceembodiment records a time at which a first quantum circuit completesexecution. From that time, an embodiment waits an additional period oftime for other circuits in the set to complete execution. In oneembodiment, the additional time period is the same as the time tocompletion of the first circuit. In another embodiment, the additionaltime period is the twice the time to completion of the first circuit. Inone embodiment, the additional time period is three times the time tocompletion of the first circuit. Other embodiments are configurable touse an adjustable time period, such as five times or ten times the timeto completion of the first circuit, or an absolute time period (e.g. tenseconds, or one minute).

An embodiment considers any circuit that completes execution within theadditional period of time as a potential solution to the problem.Because any circuits that are still executing at the end of theadditional period of time are not potential solutions to the problem, anembodiment stops execution of these circuits. An embodiment selects thelargest value of z that corresponds to a circuit in the group ofsolutions. The embodiment makes the new z the starting point, andattempts to increase z once again in a manner described herein. Thus, arace embodiment proceeds to increase z as much as possible in each stepuntil the embodiment reaches a goal value of z.

Another embodiment of a pseudo-validation method uses comparison of thecomputed variational cost function for each of the candidate points andtheir respective extrapolated parameters.

When an embodiment reaches the goal value of the continuous parameter z,having solved the hard problem, it terminates and returns the problemsolution. If an embodiment fails to reach the goal value of z, theembodiment returns the value of z that was reachable, and the solutionto the corresponding level of complexity of the problem.

An embodiment described herein provides a method to solve chemistry,machine learning, and optimization problems using adiabatic progression,but without having to perform the expensive trotterization orhamiltonian exponentiation steps which make QAOA or adiabatic simulationuntenable for Noisy Intermediate Scale Quantum (NISQ) devices. Anembodiment produces very square circuits, which are optimally suited forNISQ devices. An embodiment controls the exact pace at which to progressthe difficulty of the problem.

The manner of adiabatic progression with intermediate re-optimization tosolve hard variational quantum problems in quantum computing describedherein is unavailable in the presently available methods in thetechnological field of endeavor pertaining to quantum computing,particularly to operating a quantum data processing environment to solvehard variational problems. A method of an embodiment described herein,when implemented to execute on a device or data processing system,comprises substantial advancement of the functionality of that device ordata processing system in adiabatic progression with intermediatere-optimization to solve hard variational quantum problems in quantumcomputing.

The illustrative embodiments are described with respect to certain typesof algorithms, libraries, code, instructions, dimensions, data, devices,data processing systems, environments, components, and applications onlyas examples. Any specific manifestations of these and other similarartifacts are not intended to be limiting to the invention. Any suitablemanifestation of these and other similar artifacts can be selectedwithin the scope of the illustrative embodiments.

Furthermore, the illustrative embodiments may be implemented withrespect to any type of data, data source, or access to a data sourceover a data network. Any type of data storage device may provide thedata to an embodiment of the invention, either locally at a dataprocessing system or over a data network, within the scope of theinvention. Where an embodiment is described using a mobile device, anytype of data storage device suitable for use with the mobile device mayprovide the data to such embodiment, either locally at the mobile deviceor over a data network, within the scope of the illustrativeembodiments.

The illustrative embodiments are described using specific code, designs,architectures, protocols, layouts, schematics, and tools only asexamples and are not limiting to the illustrative embodiments.Furthermore, the illustrative embodiments are described in someinstances using particular software, tools, and data processingenvironments only as an example for the clarity of the description. Theillustrative embodiments may be used in conjunction with othercomparable or similarly purposed structures, systems, applications, orarchitectures. For example, other comparable mobile devices, structures,systems, applications, or architectures therefor, may be used inconjunction with such embodiment of the invention within the scope ofthe invention. An illustrative embodiment may be implemented inhardware, software, or a combination thereof.

The examples in this disclosure are used only for the clarity of thedescription and are not limiting to the illustrative embodiments.Additional data, operations, actions, tasks, activities, andmanipulations will be conceivable from this disclosure and the same arecontemplated within the scope of the illustrative embodiments.

Any advantages listed herein are only examples and are not intended tobe limiting to the illustrative embodiments. Additional or differentadvantages may be realized by specific illustrative embodiments.Furthermore, a particular illustrative embodiment may have some, all, ornone of the advantages listed above.

With reference to the figures and in particular with reference to FIGS.1 and 2, these figures are example diagrams of data processingenvironments in which illustrative embodiments may be implemented. FIGS.1 and 2 are only examples and are not intended to assert or imply anylimitation with regard to the environments in which differentembodiments may be implemented. A particular implementation may makemany modifications to the depicted environments based on the followingdescription.

FIG. 1 depicts a block diagram of a network of data processing systemsin which illustrative embodiments may be implemented. Data processingenvironment 100 is a network of classical computers in which theillustrative embodiments may be implemented. Data processing environment100 includes network 102. Network 102 is the medium used to providecommunications links between various devices and computers connectedtogether within data processing environment 100. Network 102 may includeconnections, such as wire, wireless communication links, or fiber opticcables.

Clients or servers are only example roles of certain data processingsystems connected to network 102 and are not intended to exclude otherconfigurations or roles for these data processing systems. Server 104and server 106 are classical data processing systems and couple tonetwork 102 along with storage unit 108. Software applications mayexecute on any computer in data processing environment 100. Clients 110,112, and 114 are also coupled to network 102. A data processing system,such as server 104 or 106, or client 110, 112, or 114 may contain dataand may have software applications or software tools executing thereon.

Only as an example, and without implying any limitation to sucharchitecture, FIG. 1 depicts certain components that are usable in anexample implementation of an embodiment. For example, servers 104 and106, and clients 110, 112, 114, are depicted as servers and clients onlyas examples and not to imply a limitation to a client-serverarchitecture. As another example, an embodiment can be distributedacross several data processing systems and a data network as shown,whereas another embodiment can be implemented on a single dataprocessing system within the scope of the illustrative embodiments. Dataprocessing systems 104, 106, 110, 112, and 114 also represent examplenodes in a cluster, partitions, and other configurations suitable forimplementing an embodiment.

Device 132 is an example of a device described herein. For example,device 132 can take the form of a smartphone, a tablet computer, alaptop computer, client 110 in a stationary or a portable form, awearable computing device, or any other suitable device. Any softwareapplication described as executing in another data processing system inFIG. 1 can be configured to execute in device 132 in a similar mannerAny data or information stored or produced in another data processingsystem in FIG. 1 can be configured to be stored or produced in device132 in a similar manner.

QCE 140 is an example of a QCE described herein. As an example, QCE 140includes CN 104, 106, and many other similar CNs 142. As an example, CNs106 and 142 may be configured as cluster 144 of CNs. QCE 140 furtherincludes one or more QCNs, such as QCN 146. A QCN, such as QCN 146,comprises one or more q-processors 148. A currently viable qubit is anexample of q-processor 148. Application 105 implements an embodimentdescribed herein. Application 105 operates on a CN, such as server 104in QCE 140. Application 105 stores an operation library, circuits, andmetadata in storage 108, or in any other suitable storage.

QCE 140 may couple to network 102 using wired connections, wirelesscommunication protocols, or other suitable data connectivity. Clients110, 112, and 114 may be, for example, personal computers or networkcomputers. Network 102 may represent a collection of networks andgateways that use the Transmission Control Protocol/Internet Protocol(TCP/IP) and other protocols to communicate with one another. FIG. 1 isintended as an example, and not as an architectural limitation for thedifferent illustrative embodiments.

Data processing environment 100 may also take the form of a cloud, andemploy a cloud computing model of service delivery for enablingconvenient, on-demand network access to a shared pool of configurablecomputing resources (e.g. networks, network bandwidth, servers,processing, memory, storage, applications, virtual machines, andservices) that can be rapidly provisioned and released with minimalmanagement effort or interaction with a provider of the service.

With reference to FIG. 2, this figure depicts a block diagram of a dataprocessing system in which illustrative embodiments may be implemented.Data processing system 200 is an example of a computer, such as servers104 and 106, or clients 110, 112, and 114 in FIG. 1, or another type ofdevice in which computer usable program code or instructionsimplementing the processes may be located for the illustrativeembodiments.

Data processing system 200 is also representative of a data processingsystem or a configuration therein, such as data processing system 132 inFIG. 1 in which computer usable program code or instructionsimplementing the processes of the illustrative embodiments may belocated. Data processing system 200 is described as a computer only asan example, without being limited thereto. Implementations in the formof other devices, such as device 132 in FIG. 1, may modify dataprocessing system 200, such as by adding a touch interface, and eveneliminate certain depicted components from data processing system 200without departing from the general description of the operations andfunctions of data processing system 200 described herein.

In the depicted example, data processing system 200 employs a hubarchitecture including North Bridge and memory controller hub (NB/MCH)202 and South Bridge and input/output (I/O) controller hub (SB/ICH) 204.Processing unit 206, main memory 208, and graphics processor 210 arecoupled to North Bridge and memory controller hub (NB/MCH) 202.Processing unit 206 may contain one or more processors and may beimplemented using one or more heterogeneous processor systems.Processing unit 206 may be a multi-core processor. Graphics processor210 may be coupled to NB/MCH 202 through an accelerated graphics port(AGP) in certain implementations.

In the depicted example, local area network (LAN) adapter 212 is coupledto South Bridge and I/O controller hub (SB/ICH) 204. Audio adapter 216,keyboard and mouse adapter 220, modem 222, read only memory (ROM) 224,universal serial bus (USB) and other ports 232, and PCI/PCIe devices 234are coupled to South Bridge and I/O controller hub 204 through bus 238.Hard disk drive (HDD) or solid-state drive (SSD) 226 and CD-ROM 230 arecoupled to South Bridge and I/O controller hub 204 through bus 240.PCI/PCIe devices 234 may include, for example, Ethernet adapters, add-incards, and PC cards for notebook computers. PCI uses a card buscontroller, while PCIe does not. ROM 224 may be, for example, a flashbinary input/output system (BIOS). Hard disk drive 226 and CD-ROM 230may use, for example, an integrated drive electronics (IDE), serialadvanced technology attachment (SATA) interface, or variants such asexternal-SATA (eSATA) and micro-SATA (mSATA). A super I/O (SIO) device236 may be coupled to South Bridge and I/O controller hub (SB/ICH) 204through bus 238.

Memories, such as main memory 208, ROM 224, or flash memory (not shown),are some examples of computer usable storage devices. Hard disk drive orsolid state drive 226, CD-ROM 230, and other similarly usable devicesare some examples of computer usable storage devices including acomputer usable storage medium.

An operating system runs on processing unit 206. The operating systemcoordinates and provides control of various components within dataprocessing system 200 in FIG. 2. The operating system may be acommercially available operating system for any type of computingplatform, including but not limited to server systems, personalcomputers, and mobile devices. An object oriented or other type ofprogramming system may operate in conjunction with the operating systemand provide calls to the operating system from programs or applicationsexecuting on data processing system 200.

Instructions for the operating system, the object-oriented programmingsystem, and applications or programs, such as application 105 in FIG. 1,are located on storage devices, such as in the form of code 226A on harddisk drive 226, and may be loaded into at least one of one or morememories, such as main memory 208, for execution by processing unit 206.The processes of the illustrative embodiments may be performed byprocessing unit 206 using computer implemented instructions, which maybe located in a memory, such as, for example, main memory 208, read onlymemory 224, or in one or more peripheral devices.

Furthermore, in one case, code 226A may be downloaded over network 201Afrom remote system 201B, where similar code 201C is stored on a storagedevice 201D. in another case, code 226A may be downloaded over network201A to remote system 201B, where downloaded code 201C is stored on astorage device 201D.

The hardware in FIGS. 1-2 may vary depending on the implementation.Other internal hardware or peripheral devices, such as flash memory,equivalent non-volatile memory, or optical disk drives and the like, maybe used in addition to or in place of the hardware depicted in FIGS.1-2. In addition, the processes of the illustrative embodiments may beapplied to a multiprocessor data processing system.

In some illustrative examples, data processing system 200 may be apersonal digital assistant (PDA), which is generally configured withflash memory to provide non-volatile memory for storing operating systemfiles and/or user-generated data. A bus system may comprise one or morebuses, such as a system bus, an I/O bus, and a PCI bus. Of course, thebus system may be implemented using any type of communications fabric orarchitecture that provides for a transfer of data between differentcomponents or devices attached to the fabric or architecture.

A communications unit may include one or more devices used to transmitand receive data, such as a modem or a network adapter. A memory may be,for example, main memory 208 or a cache, such as the cache found inNorth Bridge and memory controller hub 202. A processing unit mayinclude one or more processors or CPUs.

The depicted examples in FIGS. 1-2 and above-described examples are notmeant to imply architectural limitations. For example, data processingsystem 200 also may be a tablet computer, laptop computer, or telephonedevice in addition to taking the form of a mobile or wearable device.

Where a computer or data processing system is described as a virtualmachine, a virtual device, or a virtual component, the virtual machine,virtual device, or the virtual component operates in the manner of dataprocessing system 200 using virtualized manifestation of some or allcomponents depicted in data processing system 200. For example, in avirtual machine, virtual device, or virtual component, processing unit206 is manifested as a virtualized instance of all or some number ofhardware processing units 206 available in a host data processingsystem, main memory 208 is manifested as a virtualized instance of allor some portion of main memory 208 that may be available in the hostdata processing system, and disk 226 is manifested as a virtualizedinstance of all or some portion of disk 226 that may be available in thehost data processing system. The host data processing system in suchcases is represented by data processing system 200.

With reference to FIG. 3, this figure depicts a block diagram of anexample configuration for adiabatic progression with intermediatere-optimization to solve hard variational quantum problems in quantumcomputing, in accordance with an illustrative embodiment. Cloud 300 isan example of QCE 140 in FIG. 1. Classical computing environment 310 isan example of CCN 104 in FIG. 1. Quantum computing environment 320 is anexample of QCN 146 in FIG. 1. Application 312 is an examples ofapplication 105 in FIG. 1 and executes in server 104 in FIG. 1, or anyother suitable device in FIG. 1.

Within classical computing environment 310, application 312 receives aparametrized problem, with a single continuous parameter z, as an input.Application 312 generates quantum circuits corresponding to variousvalues of z.

Then, within quantum computing environment 320, quantum processor 340,including qubits 344 and readout circuit 346, executes a quantumcircuit. Quantum computing environment 320 also includes quantumprocessor support module 330, which outputs the results of executing thequantum circuit as a solution to the original input problem description.

With reference to FIG. 4, this figure depicts a block diagram of anexample configuration for adiabatic progression with intermediatere-optimization to solve hard variational quantum problems in quantumcomputing, in accordance with an illustrative embodiment. Application312 is the same as application 312 in FIG. 3.

Application 312 receives, as input, a known easy problem. The easyproblem has a corresponding continuous parameter z. Application 312 alsoreceives, as input, a goal value of z, representing the desiredcomplexity of the problem to be solved.

Step generator 410 generates one or more increases in the continuousparameter z. When implementing a rewind embodiment, module 410 startswith the continuous parameter z in the easy state, and increases z tothe goal state (i.e. the maximum z). If the increase in z proves to betoo large, module 410 selects a smaller increase in z from the startingpoint. If the new z succeeds in solving the problem, module 410 makesthe new z the starting point, and increases z once again to the goalstate or to another intermediate state between the new starting point orthe goal state. Module 410 continues to increase z as much as possiblein each step until application 312 generates a correct quantum circuitconfiguration with a goal value of z.

When implementing a race embodiment, module 410 starts with thecontinuous parameter z in the easy state, and generates a set ofincreases in z from the starting point. Once a group of correspondingquantum circuits completes execution, module 410 selects the largestvalue of z that corresponds to a circuit in the group of solutions.Module 410 makes the new z the starting point, and generates a new setof increases in z. Module 410 continues to increase z as much aspossible in each step until application 312 generates a correct quantumcircuit configuration with a goal value of z.

Parameter set generator 420 generates a set of variational parameterscorresponding to each increase in z generated by step generator 410. Theset of variational parameters defines a configuration of a quantumcircuit that can be used to solve the problem.

Problem solver interface module 430 takes, as input, a quantum circuitconfiguration defined by the set of variational parameters. Module 430passes the quantum circuit configuration on to quantum computingenvironment 320 for execution, and receives execution results back fromquantum computing environment 320. Module 430 also monitors an executiontime and completion time for a quantum circuit configuration.

Solution validator interface module 440 compares the output of thequantum circuit to a known correct solution to the problem obtained byanother means (such as by using some form of polynomial-time validationof the solution, or by comparing to an exact classical solution).

With reference to FIG. 5, this figure depicts an example of a hardvariational quantum problem suitable for solving using adiabaticprogression with intermediate re-optimization to solve hard variationalquantum problems in quantum computing, in accordance with anillustrative embodiment.

Graph 510 depicts a dissociation curve for a molecule of fluorine. Eachfluorine molecule consists of two fluorine atoms. The x-axis of graph510 is the distance between the two atoms, and the y-axis of graph 510is an amount of energy. Thus, graph 510 plots an amount of energy neededto pull the two atoms apart when the atoms are at a particular distancefrom each other. Computing each point in a dissociation curve is avariational problem of a particular complexity.

Graph 520 corresponds to graph 510, and depicts two error measures,total spin and total energy compared to a classical calculation,associated with computing graph 510 using conventional methods. Inparticular, area 530 indicates an area where errors in computing pointson the dissociation curve spike above a baseline. Thus, a point on thex-axis of the dissociation curve to the left of area 530 corresponds toan easier problem to solve. A point on the x-axis of the dissociationcurve to the right of area 530 corresponds to a relatively harderproblem to solve. A point on the x-axis of the dissociation curve withinarea 530 corresponds to the hardest problem to solve. If the problem ofcomputing a dissociation curve is parametrized as a variational problem,points left of area 530 on the x-axis have a low value of z, pointsright of area 530 have a mid-scale value of z, and points within area530 have a high value of z, where values of z are on a common scale(e.g. 0-1). A dissociation curve for fluorine is a non-limiting example.Many other dissociation curves have similar characteristics, and solvingfor points on such curves presents similar amounts of comparativecomputational complexity.

With reference to FIG. 6, this figure depicts an example of solving amax cut problem using adiabatic progression with intermediatere-optimization, in accordance with an illustrative embodiment. Theproblem can be solved using application 312 in FIG. 3, in a mannerdescribed herein.

As depicted, each iteration represents a location for a value of z, acontinuous parameter parametrizing an optimization problem on a graph,such as max cut, from 0 to 1. With z=0 the graph has one edge (and theproblem is easy to solve), and with z=1 the graph has all the edges inthe original problem instance.

In iteration 600, application 312, using the rewind method, performsstep 0, starting with the continuous parameter z in the easy state (0),and increasing z to the goal state (i.e. the maximum z, or 1). Theapplication then uses the quantum circuit configuration defined by theset of variational parameters to attempt to solve the problem. Theapplication compares the output of the quantum circuit to a knowncorrect solution to the problem obtained by another means, anddetermines that the step 0 error is 5.16*10⁻⁷, above a predefinedthreshold difference from the known-correct solution. Thus, theapplication concludes that step 0, increasing z from 0 to 1, was toolarge.

In iteration 610, the application performs step 1, selecting a smallerincrease in z from the starting point—here, half the size of theprevious increase, or 0.5. The application then uses the quantum circuitconfiguration defined by the set of variational parameters to attempt tosolve the problem. The application compares the output of the quantumcircuit to a known correct solution to the problem obtained by anothermeans, and determines that the step 1 error is 3.42*10⁻⁸, still above apredefined threshold difference from the known-correct solution. Thus,the application concludes that step 1, increasing z from 0 to 0.5, wasalso too large.

In iteration 620, the application performs step 2, selecting an evensmaller increase in z from the starting point—here, half the size of theprevious increase, or 0.25. The application then uses the quantumcircuit configuration defined by the set of variational parameters toattempt to solve the problem. This time, the step 2 error is 4.56*10⁻⁹,below the predefined threshold difference from the known-correctsolution.

As a result, in iteration 630, the application performs step 3,increasing z from 0.25 to 0.5, and obtaining the result of thecorresponding quantum circuit configuration. This time, the step 3 erroris 3.59*10⁻⁹, still below the predefined threshold difference from theknown-correct solution.

As a result, in iteration 640, the application performs step 4,increasing z from 0.5 to 1, and obtaining the result of thecorresponding quantum circuit configuration. This time, the step 4 erroris 2.23*10⁻⁹, also below the predefined threshold difference from theknown-correct solution. Because the error is below the thresholddifference and z=1, the goal state of the problem has been solved.

The increases depicted in FIG. 6 do not depict additional steps that theapplication could have performed to reach the goal state. An applicationcould have performed additional steps, fewer steps, or increases ofdifferent sizes without departing from the scope of the illustrativeembodiments.

With reference to FIG. 7, this figure depicts an example of computing adissociation curve using adiabatic progression with intermediatere-optimization, in accordance with an illustrative embodiment. Pointson the curve can be solved using application 312 in FIG. 3, in a mannerdescribed herein.

Graph 700 depicts points on a dissociation curve obtained usingapplication 312. The x-axis of graph 700 denotes interatomic distance,measured in Angstroms, and the y-axis depicts dissociation energies.Graph 700 also depicts ideal dissociation curve 710, obtained usinganother method.

Area 720 depicts a portion of the x-axis known to have highercomplexity, parametrized by a high value of z (e.g. close to 1 on a 0-1scale). As a result, the application begins solving for points on thecurve further to the left on the x-axis.

In particular, the application begins with point 730, which has beensolved. In step 740, the application, using the rewind method, increasesz to attempt to solve for a point having a x value higher than that ofpoint 730. However, the application's initial solution is above athreshold difference from the point on curve 710 having the same xvalue.

As a result, in step 750 the application generates a smaller increase inz than that of step 740. The smaller increase results in a solutionbelow a threshold difference from the point on curve 710 having the samex value.

As a result, the z value of step 750 becomes the new starting point. Instep 760 the application once again increases z, to the same level aswas attempted in step 740. Step 750 also results in a solution below athreshold difference from the point on curve 710 having the same xvalue. The application continues in a manner described herein until allthe desired points on the dissociation curve have been solved for.

With reference to FIG. 8, this figure depicts another example ofcomputing a dissociation curve using adiabatic progression withintermediate re-optimization, in accordance with an illustrativeembodiment. Points on the curve can be solved using application 312 inFIG. 3, in a manner described herein. Graph 700, curve 710, and point730 are the same as graph 700, curve 710, and point 730 in FIG. 7.

The application begins with point 730, which has been solved. Theapplication, using the race method, generates a set of increases in z toattempt to solve for points having x values higher than that of point730. The application generates quantum circuits corresponding to the setof increases in z, and waits a predetermined time period for the quantumcircuits to complete executing. At the end of the time period, all ofthe quantum circuits corresponding to area 840 have completed executing.The quantum circuits corresponding to points further to the right alongthe x-axis, for example, points 860 and 870, have not completedexecuting. Because point 850 represents the highest increase in z frompoint 730 corresponding to a circuit within area 840, the applicationselects the z value corresponding to point 850 as the new starting pointand repeats the process. The application continues, in a mannerdescribed herein, until all desired points on the dissociation curvehave been solved for.

With reference to FIG. 9, this figure depicts a flowchart of an exampleprocess for adiabatic progression with intermediate re-optimization tosolve hard variational quantum problems in quantum computing inaccordance with an illustrative embodiment. Process 900 can beimplemented in application 312 in FIG. 3 or FIG. 4, and illustratesoperation of a rewind method described herein.

In block 902, the application generates an increase in a continuousparameter of a parametrized optimization problem, where a current statein the continuous parameter corresponds to a solvable state of theproblem. In block 904, the application generates a set of variationalparameters corresponding to the increase. In block 906, the applicationuses the quantum circuit configuration specified by the set ofvariational parameters to solve the problem. In block 908, theapplication checks whether the problem solution is more than a thresholddifferent from a known correct solution to the problem. If so (“YES”path of block 908), in block 910 the application generates anotherincrease, smaller than the previous increase, in the continuousparameter of the problem, and then returns to block 904. Otherwise (“NO”path of block 908), in block 912 the application checks whether thecontinuous parameter z is at the goal state. If yes, the applicationends. Otherwise (“NO” path of block 912), the application returns toblock 902.

With reference to FIG. 10, this figure depicts a flowchart of anotherexample process for adiabatic progression with intermediatere-optimization to solve hard variational quantum problems in quantumcomputing in accordance with an illustrative embodiment. Process 1000can be implemented in application 312 in FIG. 3 or FIG. 4, andillustrates operation of a race method described herein.

In block 1002, the application generates a set of different-sizedincreases in a continuous parameter of a parametrized optimizationproblem, where a current state in the continuous parameter correspondsto a solvable state of the problem. In block 1004, the applicationgenerates a set of variational parameters corresponding to eachincrease. In block 1006, the application uses the quantum circuitconfiguration specified by each set of variational parameters to solvethe problem. In block 1008, the application waits for a predeterminedtime period after one of the quantum circuits completes executing. Inblock 1010 the application selects the quantum circuit that correspondsto the largest increase in the continuous parameter that also completedexecution within the time period, and replaces the current value of zwith that largest increase. In block 1012 the application checks whetherthe continuous parameter z is at the goal state. If yes, the applicationends. Otherwise (“NO” path of block 1012), the application returns toblock 1002.

Thus, a computer implemented method, system or apparatus, and computerprogram product are provided in the illustrative embodiments foradiabatic progression with intermediate re-optimization to solve hardvariational quantum problems in quantum computing and other relatedfeatures, functions, or operations. Where an embodiment or a portionthereof is described with respect to a type of device, the computerimplemented method, system or apparatus, the computer program product,or a portion thereof, are adapted or configured for use with a suitableand comparable manifestation of that type of device.

Where an embodiment is described as implemented in an application, thedelivery of the application in a Software as a Service (SaaS) model iscontemplated within the scope of the illustrative embodiments. In a SaaSmodel, the capability of the application implementing an embodiment isprovided to a user by executing the application in a cloudinfrastructure. The user can access the application using a variety ofclient devices through a thin client interface such as a web browser(e.g., web-based e-mail), or other light-weight client-applications. Theuser does not manage or control the underlying cloud infrastructureincluding the network, servers, operating systems, or the storage of thecloud infrastructure. In some cases, the user may not even manage orcontrol the capabilities of the SaaS application. In some other cases,the SaaS implementation of the application may permit a possibleexception of limited user-specific application configuration settings.

The present invention may be a system, a method, and/or a computerprogram product at any possible technical detail level of integration.The computer program product may include a computer readable storagemedium (or media) having computer readable program instructions thereonfor causing a processor to carry out aspects of the present invention.

The computer readable storage medium can be a tangible device that canretain and store instructions for use by an instruction executiondevice. The computer readable storage medium may be, for example, but isnot limited to, an electronic storage device, a magnetic storage device,an optical storage device, an electromagnetic storage device, asemiconductor storage device, or any suitable combination of theforegoing. A non-exhaustive list of more specific examples of thecomputer readable storage medium includes the following: a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (ROM), an erasable programmable read-only memory (EPROMor Flash memory), a static random access memory (SRAM), a portablecompact disc read-only memory (CD-ROM), a digital versatile disk (DVD),a memory stick, a floppy disk, a mechanically encoded device such aspunch-cards or raised structures in a groove having instructionsrecorded thereon, and any suitable combination of the foregoing. Acomputer readable storage medium, including but not limited tocomputer-readable storage devices as used herein, is not to be construedas being transitory signals per se, such as radio waves or other freelypropagating electromagnetic waves, electromagnetic waves propagatingthrough a waveguide or other transmission media (e.g., light pulsespassing through a fiber-optic cable), or electrical signals transmittedthrough a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network may comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device.

Computer readable program instructions for carrying out operations ofthe present invention may be assembler instructions,instruction-set-architecture (ISA) instructions, machine instructions,machine dependent instructions, microcode, firmware instructions,state-setting data, configuration data for integrated circuitry, oreither source code or object code written in any combination of one ormore programming languages, including an object oriented programminglanguage such as Smalltalk, C++, or the like, and procedural programminglanguages, such as the “C” programming language or similar programminglanguages. The computer readable program instructions may executeentirely on the user's computer, partly on the user's computer, as astand-alone software package, partly on the user's computer and partlyon a remote computer or entirely on the remote computer or server. Inthe latter scenario, the remote computer may be connected to the user'scomputer through any type of network, including a local area network(LAN) or a wide area network (WAN), or the connection may be made to anexternal computer (for example, through the Internet using an InternetService Provider). In some embodiments, electronic circuitry including,for example, programmable logic circuitry, field-programmable gatearrays (FPGA), or programmable logic arrays (PLA) may execute thecomputer readable program instructions by utilizing state information ofthe computer readable program instructions to personalize the electroniccircuitry, in order to perform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions.

These computer readable program instructions may be provided to aprocessor of a general purpose computer, special purpose computer, orother programmable data processing apparatus to produce a machine, suchthat the instructions, which execute via the processor of the computeror other programmable data processing apparatus, create means forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks. These computer readable program instructionsmay also be stored in a computer readable storage medium that can directa computer, a programmable data processing apparatus, and/or otherdevices to function in a particular manner, such that the computerreadable storage medium having instructions stored therein comprises anarticle of manufacture including instructions which implement aspects ofthe function/act specified in the flowchart and/or block diagram blockor blocks.

The computer readable program instructions may also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational steps to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the blocks may occur out of theorder noted in the Figures. For example, two blocks shown in successionmay, in fact, be executed substantially concurrently, or the blocks maysometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

Embodiments of the present invention may also be delivered as part of aservice engagement with a client corporation, nonprofit organization,government entity, internal organizational structure, or the like.Aspects of these embodiments may include configuring a computer systemto perform, and deploying software, hardware, and web services thatimplement, some or all of the methods described herein. Aspects of theseembodiments may also include analyzing the client's operations, creatingrecommendations responsive to the analysis, building systems thatimplement portions of the recommendations, integrating the systems intoexisting processes and infrastructure, metering use of the systems,allocating expenses to users of the systems, and billing for use of thesystems. Although the above embodiments of present invention each havebeen described by stating their individual advantages, respectively,present invention is not limited to a particular combination thereof. Tothe contrary, such embodiments may also be combined in any way andnumber according to the intended deployment of present invention withoutlosing their beneficial effects.

What is claimed is:
 1. A computer-implemented method comprising:generating a first set of increases from a current state of a continuousparameter of a parametrized optimization problem, wherein the currentstate of the continuous parameter corresponds to a first difficultylevel of the parametrized optimization problem; generating a set ofvariational parameters corresponding to each of the first set ofincreases, each set of variational parameters specifying a configurationof a quantum circuit in a first set of configurations of the quantumcircuit; causing a hybrid classical-quantum computing device to executeeach configuration in the first set of configurations, execution of thefirst set of configurations resulting in a first set of outputs; andcausing, responsive to determining that a sum of the current state ofthe continuous parameter and a largest increase in the first set ofincreases in the continuous parameter is less than or equal to a goalstate of the continuous parameter, the hybrid classical-quantumcomputing device to execute a second set of configurations of thequantum circuit, the second set of configurations of the quantum circuitspecified by each of a second set of variational parameters generatedfrom a second set of increases in the continuous parameter from thecurrent state of the continuous parameter, wherein the goal state of thecontinuous parameter corresponds to a second difficulty level of theparametrized optimization problem higher than the first difficultylevel.
 2. The computer-implemented method of claim 1, wherein the hybridclassical-quantum computing device comprises a Noisy Intermediate ScaleQuantum (NISQ) device.
 3. The computer-implemented method of claim 2,wherein the first set of configurations of the quantum circuit eachcomprises a square quantum circuit, the square quantum circuit beingsuited for implementing on the NISQ device.
 4. The computer-implementedmethod of claim 1, further comprising: controlling, according to thevalues of the first set of increases in the continuous parameter, a paceat which to progress a difficulty level of the parametrized optimizationproblem.
 5. The computer-implemented method of claim 4, wherein thelargest increase in the first set of increases comprises a differencebetween the goal state of the continuous parameter and the current stateof the continuous parameter.
 6. The computer-implemented method of claim4, wherein the largest increase in the first set of increases comprisesa difference between the goal state of the continuous parameter and thecurrent state of the continuous parameter, the difference then dividedby n, where n is a positive whole number.
 7. The computer-implementedmethod of claim 4, wherein the first set of increases comprises a set ofequally-spaced increases between the current state of the continuousparameter and the goal state of the continuous parameter.
 8. Thecomputer-implemented method of claim 4, wherein the largest increase infirst set of increases to a configuration of the quantum circuit thatcompleted execution within an execution time period.
 9. A computerusable program product comprising one or more computer-readable storagedevices, and program instructions stored on at least one of the one ormore storage devices, the stored program instructions comprising:program instructions to generate a first set of increases from a currentstate of a continuous parameter of a parametrized optimization problem,wherein the current state of the continuous parameter corresponds to afirst difficulty level of the parametrized optimization problem; programinstructions to generate a set of variational parameters correspondingto each of the first set of increases, each set of variationalparameters specifying a configuration of a quantum circuit in a firstset of configurations of the quantum circuit; program instructions tocause a hybrid classical-quantum computing device to execute eachconfiguration in the first set of configurations, execution of the firstset of configurations resulting in a first set of outputs; and programinstructions to cause, responsive to determining that a sum of thecurrent state of the continuous parameter and a largest increase in thefirst set of increases in the continuous parameter is less than or equalto a goal state of the continuous parameter, the hybridclassical-quantum computing device to execute a second set ofconfigurations of the quantum circuit, the second set of configurationsof the quantum circuit specified by each of a second set of variationalparameters generated from a second set of increases in the continuousparameter from the current state of the continuous parameter, whereinthe goal state of the continuous parameter corresponds to a seconddifficulty level of the parametrized optimization problem higher thanthe first difficulty level.
 10. The computer usable program product ofclaim 9, wherein the hybrid classical-quantum computing device comprisesa Noisy Intermediate Scale Quantum (NISQ) device.
 11. The computerusable program product of claim 10, wherein the first set ofconfigurations of the quantum circuit each comprises a square quantumcircuit, the square quantum circuit being suited for implementing on theNISQ device.
 12. The computer usable program product of claim 9, furthercomprising: program instructions to control, according to the values ofthe first set of increases in the continuous parameter, a pace at whichto progress a difficulty level of the parametrized optimization problem.13. The computer usable program product of claim 12, wherein the largestincrease in the first set of increases comprises a difference betweenthe goal state of the continuous parameter and the current state of thecontinuous parameter.
 14. The computer usable program product of claim12, wherein the largest increase in the first set of increases comprisesa difference between the goal state of the continuous parameter and thecurrent state of the continuous parameter, the difference then dividedby n, where n is a positive whole number.
 15. The computer usableprogram product of claim 12, wherein the first set of increasescomprises a set of equally-spaced increases between the current state ofthe continuous parameter and the goal state of the continuous parameter.16. The computer usable program product of claim 12, wherein the largestincrease in first set of increases to a configuration of the quantumcircuit that completed execution within an execution time period. 17.The computer usable program product of claim 9, wherein the computerusable code is stored in a computer readable storage device in a dataprocessing system, and wherein the computer usable code is transferredover a network from a remote data processing system.
 18. The computerusable program product of claim 9, wherein the computer usable code isstored in a computer readable storage device in a server data processingsystem, and wherein the computer usable code is downloaded over anetwork to a remote data processing system for use in a computerreadable storage device associated with the remote data processingsystem, further comprising: program instructions to meter use of thecomputer usable code associated with the request; and programinstructions to generate an invoice based on the metered use.
 19. Acomputer system comprising one or more processors, one or morecomputer-readable memories, and one or more computer-readable storagedevices, and program instructions stored on at least one of the one ormore storage devices for execution by at least one of the one or moreprocessors via at least one of the one or more memories, the storedprogram instructions comprising: program instructions to generate afirst set of increases from a current state of a continuous parameter ofa parametrized optimization problem, wherein the current state of thecontinuous parameter corresponds to a first difficulty level of theparametrized optimization problem; program instructions to generate aset of variational parameters corresponding to each of the first set ofincreases, each set of variational parameters specifying a configurationof a quantum circuit in a first set of configurations of the quantumcircuit; program instructions to cause a hybrid classical-quantumcomputing device to execute each configuration in the first set ofconfigurations, execution of the first set of configurations resultingin a first set of outputs; and program instructions to cause, responsiveto determining that a sum of the current state of the continuousparameter and a largest increase in the first set of increases in thecontinuous parameter is less than or equal to a goal state of thecontinuous parameter, the hybrid classical-quantum computing device toexecute a second set of configurations of the quantum circuit, thesecond set of configurations of the quantum circuit specified by each ofa second set of variational parameters generated from a second set ofincreases in the continuous parameter from the current state of thecontinuous parameter, wherein the goal state of the continuous parametercorresponds to a second difficulty level of the parametrizedoptimization problem higher than the first difficulty level.
 20. Thecomputer system of claim 19, wherein the hybrid classical-quantumcomputing device comprises a Noisy Intermediate Scale Quantum (NISQ)device.
 21. The computer system of claim 20, wherein the first set ofconfigurations of the quantum circuit each comprises a square quantumcircuit, the square quantum circuit being suited for implementing on theNIS Q device.
 22. The computer system of claim 19, further comprising:program instructions to control, according to the values of the firstset of increases in the continuous parameter, a pace at which toprogress a difficulty level of the parametrized optimization problem.23. The computer system of claim 22, wherein the largest increase in thefirst set of increases comprises a difference between the goal state ofthe continuous parameter and the current state of the continuousparameter.
 24. The computer system of claim 22, wherein the largestincrease in the first set of increases comprises a difference betweenthe goal state of the continuous parameter and the current state of thecontinuous parameter, the difference then divided by n, where n is apositive whole number.
 25. The computer system of claim 22, wherein thefirst set of increases comprises a set of equally-spaced increasesbetween the current state of the continuous parameter and the goal stateof the continuous parameter.